How to make a beveled cutout in a sphere?
For natural and smooth indentation you could use Displace modifier with a texture. Here we have a rotated UV sphere that has UV coordinates projected from view. The white parts of the texture are moved along the surface normal by the specified amount, negative numbers making an indentation.

Steps to recreate this:


Add an UV sphere, enable smooth shading and rotate it 90 degrees around y-axis
Enter edit mode and select all
Enter orthogonal top down view and press U > Project from view
Exit edit mode and add subsurface modifier, set it to 4 view and render
Add a displace modifier, set the strength to -0.05 and click to create new texture
Go to Texture tab, select you newly created texture from the drop-down and set its type to image or movie
Create and paint new image, open one from file or browser the drop-down for existing image to be used with the texture. Note the image should have soft gradient from white areas to black areas, this will affect the displace strength accordingly.
If you need to edit the placement of the displace texture, go into edit mode and find the UV layout in an image editor window. In the image editor window you can select all your UVs and move, rotate or scale them to align the texture.




This approach often requires a high number of vertices for non-jagged results but it's faster than trying to make the appropriate geometry especially if the shape of the indentation is a bit uncertain.

Alternative approach might be to model the shape on a flat surface geometry and curve the flat surface on a sphere using shrinkwrap or lattice.